The fourth dimension

RECAPITULATION AND EXTENSION 217

In an analogous manner, in four dimensions we can have rotation round a curved plane, if I may use the expression. A sphere can be turned inside out in four dimensions.

Let fig. 11 represent a

z spherical surface, on each

side of which a layer of

f matter exists. The thick-

ness of the matter is represented by the rods cp and

EF, extending equally withY out and within.

Now, take the section of the sphere by the yz plane we have a circle—fig. 12. Now, let the waxis be drawn in place of the x axis so that we have the space of yzw

represented. In this space all that there will be seen ot the sphere is the circle drawn.

Here we see that there is no obstacle to prevent the

z rods turning round. If

the matter is so elastic

Fr that it will give enough

o for the particles at E and

C to be separated as they are at F and D, they can rotate round to the position D and F, and a similar motion is possible for all other particles. There is no matter or obstacle to prevent them Fig. 12 (140), from moving out in the

w direction, and then on round the circumference as an axis. Now, what will hold for one section will hold for

Fig. 11 (139).